is a left Quillen functor for every \(F \in {{K}}\) .
It then follows from Proposition REF  and the arguments in the proof of Proposition REF  that the natural transformation (REF ) is a natural weak equivalence of functors \({{K}}_\infty \rightarrow {{K}}_\infty ^p\) .
Finally, we use that the functor \(B((-),{C},{{Y}})\)  agrees with Dugger's cofibrant replacement functor \(Q^p\)  for \({{K}}_\infty ^p\)  from [1]}.
In particular, it comes with a natural weak equivalence \(q^p \colon Q^p \rightarrow 1_{{{K}}_\infty }\) .
Composing \(q^p\)  with the morphism (REF ), we obtain a natural weak equivalence \(q^{\prime } \colon Q^{\prime } \rightarrow 1_{{{K}}_\infty }\) .
Putting everything together, we have proven
